adding more text to a query string

You can try and use set_query_var() very very early, before the query is parsed and queried for, for example during the pre_get_posts action. Although thinking about it, if you’re using pre_get_posts it’s likely easier to do something like this (off the top of my head):

add_action( 'pre_get_posts', function( $query ) {
    if ( ! $query->is_main_query() || ! $query->is_search() )
        return;

    $query->set( 's', $query->get( 's' ) . sprintf( ' %s %s', $_GET['s1'], $_GET['s2'] ) );
});